Method and device for carrier-frequency synchronization of a vestigial-sideband-modulated signal

ABSTRACT

A method and a device for carrier-frequency synchronization of a vestigial-sideband-modulated received signal (r VSB (t)) with a carrier signal (e j(2π(f     T     +Δf)t+Δφ) ), which is affected by a frequency offset and/or phase offset (Δf, Δφ), estimates the frequency offset and/or phase offset (Δf, Δφ) of the carrier signal (e j(2π(f     T     +Δf)t+Δφ) ) by means of a maximum-likelihood estimation. For this purpose, the vestigial-sideband-modulated received signal (r VSB (t)) is converted into a modified vestigial-sideband-modulated received signal (x VSB ′(t′)), which provides time-discrete complex rotating phasors (|x VSB ′(t′)|·e j2πΔf·t′+Δφ ), in which only the time-discrete phases (2πΔft′+Δφ) are dependent upon the frequency offset and/or phase offset (Δf, Δφ).

The invention relates to a method and a device for carrier-frequencysynchronisation with a vestigial-sideband-modulated signal (VSB).

If transmitters and receivers are synchronised with one another within atransmission system, a transmitter-end adaptation and a receiver-endadaptation of the clock and carrier signal takes place with regard tophase position and frequency respectively. The carrier-frequencysynchronisation to be considered below presupposes a received signal,which is synchronised with regard to the clock signal.

Document DE 103 09 262 A1 describes a method for carrier-frequencysynchronisation of a signal with digital symbol sequences, in which thefrequency offset and/or phase offset of the carrier signal is estimatedfrom the demodulated received signal by means of maximum-likelihoodestimation. The received signal containing the digital symbol sequencesconsists of complex rotating phasors associated with the individualsampling times, of which the discrete phases depend only on the soughtfrequency offset and/or phase offset of the carrier signal, and of whichthe discrete amplitudes depend only on the digital symbol values of thedemodulated received signal. The maximum-likelihood estimation of thesought frequency offset and/or phase offset of the carrier signal takesplace by maximising the likelihood function, which is formed from thesum of the real components of all of the time-discrete, complex rotatingphasors of the received signal. Maximising the likelihood function isachieved by rotating the associated, complex rotating phasor of thereceived signal in a clockwise direction for each sampling time at thelevel of the sought frequency offset and/or phase offset, so that it isdisposed on the real axis. In this manner, it is possible, to obtain thesought frequency offset and/or phase offset of the carrier signal byobserving the extreme-values of the likelihood function separately forthe frequency offset and/or phase offset.

The time-discrete received signal in DE 103 09 262 A1 provides onecomplex rotating phasor for each sampling time, of which the phase valuedepends only upon the frequency offset and/or phase offset of thecarrier signal, and of which the amplitude value depends on the symbolvalue of the received signal sequence to be transmitted at therespective sampling time. A time-discrete received signal of this kindis based upon a comparatively simple modulation, for example, aconventional amplitude modulation. By contrast, a VSB received signalprovides no time-discrete, complex rotating phasors, of which thetime-discrete phases are dependent only upon the frequency offset andphase offset. In this case, the use of a maximum-likelihood estimationto estimate the sought frequency offset and/or phase offset of thecarrier signal in the sense of the method and the device known from DE103 09 262 A1 therefore fails to achieve the goal.

The invention is accordingly based upon the object of providing a methodand a device for estimating the frequency offset and/or phase offset inthe carrier signal in the case of a vestigial-sideband-modulatedreceived signal using a maximum-likelihood estimation.

The object of the invention is achieved by a method forcarrier-frequency synchronisation with the features of claim 1 and by adevice for carrier-frequency synchronisation with the features of claim10. Further developments of the invention are specified in the dependentclaims.

In a first stage, the VSB received signal is converted into a modifiedVSB signal, which, in an equivalent manner to a quadrature-modulatedsignal—for example, a PAM, QPSK or π/4-QPSK signal—providestime-discrete, complex rotating phasors consisting respectively of anin-phase component and a quadrature component. The symbol duration ofthe VSB received signal according to the invention is therefore adjustedto the level of half the symbol duration of a quadrature-modulatedsignal, and the accordingly-adapted VSB received signal is converted bydown mixing into a modified VSB signal, which consists of a complexrotating phasor typical for a quadrature-modulated signal and provides asignal display equivalent to that of an offset QPSK signal.

In a second stage, this modified VSB received signal, which isequivalent to an offset QPSK signal, is additionally converted, in orderto realise time-discrete complex rotating phasors, which are dependentonly upon the frequency offset and phase offset of the carrier signal.The modified VSB signal is therefore converted by sampling with anoversampling factor of typically eight, estimation filtering with asignal-adapted estimation filter and three further signal-processingstages according to the invention, in order to realise anadditionally-modified VSB received signal, of which the time-discretecomplex rotating phasors each provide phases, which depend only upon thefrequency offset and/or phase offset of the carrier signal used.

The first signal-processing stage involves a further sampling, whichgenerates a time-discrete, modified VSB received signal with twosampling values per symbol period. This accordingly re-sampled VSBreceived signal contains in each of its discrete, complex rotatingphasors an additional phase dependent upon the respective sampling time,which, in the subsequent, second signal-processing stage, is compensatedby a respective, inverse phase, in the context of a complexmultiplication with a complex rotating phasor. In a thirdsignal-processing stage, the VSB received signal, freed from itsadditional phase in each of the time-discrete complex rotating phasors,is finally subjected to modulus-scaled squaring, in order to ensure thatthe amplitude of each time-discrete complex rotating phasor of themodified VSB received signal has a positive value.

With the method according to the invention and the device according tothe invention for carrier-frequency synchronisation, a modified VSBreceived signal, of which the time-discrete complex rotating phasorseach provide phases, which are dependent only upon the frequency offsetand/or phase offset of the carrier signal used, is therefore formed fromthe VSB received signal.

The time-discrete phases of the multi-modified, time-discrete VSBreceived signal are then determined via an argument function, and aphase characteristic is formed. This phase characteristic of themodified VSB received signal, which is periodic over the period 2·π andnon-steady, is then “steadied” at the non-steady points to provide aphase-continuous phase characteristic of the modified VSB receivedsignal.

A phase-continuous phase characteristic of a multi-modified VSB receivedsignal generated in this manner can be subjected to a maximum-likelihoodestimation in the sense of DE 103 09 262, in order to determine afrequency offset and/or phase offset possibly occurring in the carriersignal used, thereby approaching the goal of a subsequentcarrier-frequency synchronisation of the VSB received signal.

A preferred exemplary embodiment of the method according to theinvention for carrier-frequency synchronisation of a VSB signal and thedevice according to the invention for carrier-frequency synchronisationof a VSB signal are explained in greater detail below with reference tothe drawings. The drawings are as follows:

FIG. 1 shows an extended block circuit diagram of the transmissionsystem;

FIG. 2 shows a reduced block circuit diagram of the transmission system;

FIG. 3 shows a block circuit diagram of the device for carrier-frequencysynchronisation according to the invention;

FIG. 4 shows a complex phasor diagram of a modified VSB received signalaccording to the invention;

FIG. 5 shows a flow chart of the method according to the invention forcarrier-frequency synchronisation.

Before describing an embodiment of the method according to the inventionand the device according to the invention for carrier-frequencysynchronisation with a VSB received signal in greater detail withreference to FIGS. 3 to 5, the following paragraphs present a derivationof the required mathematical background.

Accordingly, in a first stage, the VSB signal s_(VSB)(t) is convertedaccording to equation (1) into a modified VSB received signal, whichprovides a signal display equivalent to an offset QPSK signal with acomplex rotating phasor.

$\begin{matrix}{{s_{VSB}(t)} = {\sum\limits_{v = {- \infty}}^{+ \infty}{{q(v)} \cdot {\delta( {t - {v \cdot T_{VSB}}} )}}}} & (1)\end{matrix}$

In this equation, the values q(v) in a 2VSB signal represent the symbolsequence with the symbol alphabet {+1+pilot, −1+pilot} and the symbolduration T_(VSB). A pilot carrier, for which the condition pilot=0applies, is conventionally contained therein. In the deliberationspresented below, ideal conditions—pilot=0—are assumed.

In a complex baseband model of a transmission system 1 fortime-continuous complex signals, as shown in FIG. 1, the VSB signalS_(VSB)(t) is supplied to a transmission filter 2. As in the case of aquadrature-modulated signal—such as a PAM, QPSK or π/4-QPSK modulatedsignal—, this is realised as a cosine filter with a transmissionfunction H_(S)(f) and a roll-off factor r, as shown in equation (2).

$\begin{matrix}{{H_{s}(f)} = \{ {\begin{matrix}{1} & {{für}{{f{{< \frac{f_{s}}{2}}}}}} \\{\cos\;\lbrack {\frac{\pi\;{{f}}}{2{r \cdot f_{s}}} - \frac{\pi\;( {1 - r} )}{4r}} \rbrack} & {{{{für}( {1 - r} )}\frac{f_{s}}{2}} < {{f{{\leq {( {1 + r} )\frac{f_{s}}{2}}}}}}} \\{0} & {{{{für}( {1 + r} )}\frac{f_{s}}{2}} < {{f}}}\end{matrix}\lbrack {{für} = {for}} \rbrack} } & (2)\end{matrix}$

By contrast with the transmission filter in a PAM, QPSK or π/4 QPSKmodulated signal, the transmission filter 2 associated with a VSB signalS_(VSB)(t) is a cosine filter symmetrical to the frequency

${f = {\frac{1}{4} \cdot f_{SVSB}}},$wherein f_(SVSB) is the symbol frequency of the VSB signal inverse tothe symbol period T_(VSB). Its transmission function H_(SVSB)(f) istherefore derived from a displacement of the transmission functionH_(S)(f) of a quadrature-modulated signal by the frequency

$f = {\frac{1}{4} \cdot f_{SVSB}}$in the sense of equation (3).

$\begin{matrix}{{H_{SVSB}(f)} = {H_{s}( {f - {\frac{1}{4} \cdot f_{SVSB}}} )}} & (3)\end{matrix}$

The context shown in equation (4) applies for the symbol rate f_(s) fromequation (2), which relates to classical, quadrature-modulated signals,and for the symbol frequency of a VSB signal f_(SVSB):

$\begin{matrix}{f_{s} = {\frac{1}{2} \cdot f_{SVSB}}} & (4)\end{matrix}$

The impulse response h_(SVSB)(t) of the transmission filter 2 for a VSBsignal is therefore derived from the following equation (5):

$\begin{matrix}{{h_{SVSB}(t)} = {{h_{s}(t)} \cdot {\mathbb{e}}^{j\frac{2\;\pi}{4T_{VSB}}t}}} & (5)\end{matrix}$

The VSB signal S_(FVSB)(t) disposed at the output of the transmissionfilter 2 is derived from a convolution of the VSB received signalS_(VSB)(t) according to equation (1) with the impulse responseh_(SVSB)(t) of the transmission filter 2 according to equation (5) andis described mathematically by equation (6), which is mathematicallyconverted over several stages:

$\begin{matrix}\begin{matrix}{{s_{FVSB}(t)} = {( {{h_{s}(t)} \cdot {\mathbb{e}}^{j\frac{2\;\pi}{4T_{VSB}}t}} ) \star {\sum\limits_{v = {- \infty}}^{+ \infty}{{q(v)} \cdot {\delta( {t - {v \cdot T_{VSB}}} )}}}}} \\{= {\sum\limits_{v = {- \infty}}^{+ \infty}{{q(v)} \cdot {h_{s}( {t - {v \cdot T_{VSB}}} )} \cdot {\mathbb{e}}^{j\frac{2\;\pi}{4T_{VSB}}{({t - {v \cdot T_{VSB}}})}}}}} \\{= {{\mathbb{e}}^{j\frac{2\;\pi}{4T_{VSB}}t} \cdot {\sum\limits_{v = {- \infty}}^{+ \infty}{{q(v)} \cdot {\mathbb{e}}^{{- j}\frac{\;\pi}{2}v} \cdot {h_{s}( {t - {v \cdot T_{VSB}}} )}}}}}\end{matrix} & (6)\end{matrix}$

According to equation (7), the value b(v) is introduced for the term

${q(v)} \cdot {\mathbb{e}}^{{- j}\frac{\;\pi}{2}v}$in equation (6):

$\begin{matrix}{{b(v)}\text{:}{= {q(v)} \cdot {\mathbb{e}}^{{- j}\frac{\;\pi}{2}v}}} & (7)\end{matrix}$

If the value b(v) is observed separately for even-numbered andodd-numbered v, then for even-numbered v=2n (n: integer), themathematical relationship of equation (8) is obtained, which, aftermathematical conversion, gives real values a_(R)(n):

$\begin{matrix}\begin{matrix}{{b(v)}{_{v = {2n}}{= {{\mathbb{e}}^{{- j}\frac{\pi}{2}2n} \cdot {q( {2n} )}}}}} \\{= {{\mathbb{e}}^{{- j}\;\pi\; n} \cdot {q( {2n} )}}} \\{= {{( {- 1} )^{n} \cdot {q( {2n} )}}\text{:} = {a_{R}(n)}}}\end{matrix} & (8)\end{matrix}$

With odd-numbered v=2n+1 (n: integer), for the value b(v), themathematical relationship of equation (9), is obtained, which, aftermathematical conversion, gives complex values j·a_(x)(n):

$\begin{matrix}\begin{matrix}{{b(v)}{_{v = {{2n} + 1}}{= {{\mathbb{e}}^{{- j}\frac{\pi}{2}{({{2n} + 1})}} \cdot {q( {{2n} + 1} )}}}}} \\{= {{j \cdot ( {- 1} )^{n + 1} \cdot {q( {{2n} + 1} )}}\text{:}}} \\{= {j \cdot {a_{I}(n)}}}\end{matrix} & (9)\end{matrix}$

The term

${q(v)} \cdot {\mathbb{e}}^{{- j}\frac{\pi}{2}v}$in equation (6) can be presented according to equation (10) foreven-numbered v=2n as the even-numbered term b(v)|_(v=2n), and forodd-numbered v=2n+1, as the odd-numbered term b(v)|_(v=2n+1):

$\begin{matrix}{{{q(v)} \cdot {\mathbb{e}}^{{- j}\frac{\pi}{2}v}} = {{b(v)}{_{v = {2n}}{+ {b(v)}}}_{v = {{2n} + 1}}}} & (10)\end{matrix}$

The mathematical relationship for the output signal S_(FVSB)(t) at theoutput of the transmission filter 2 in equation (6) can therefore beconverted according to equation (10), taking into consideration equation(8) and (9), as shown in equation (11):

$\begin{matrix}{{s_{FVSB}(t)} = {{\mathbb{e}}^{j\frac{2\;\pi}{4T_{VSB}}t} \cdot ( {{\sum\limits_{n = {- \infty}}^{+ \infty}{{a_{R}(n)} \cdot {h_{S}( {t - {2{n \cdot T_{VSB}}}} )}}} + {j \cdot {\sum\limits_{n = {- \infty}}^{+ \infty}{{a_{I}(n)} \cdot {h_{S}( {t - {2{n \cdot T_{VSB}}} - T_{{{VSB})})}} }}}}} }} & (11)\end{matrix}$

The subsequent lag element 3 models the time offset ε·T occurring as aresult of the absent or inadequate synchronisation between thetransmitter and the receiver, which is derived from a timing offset ε.The timing offset ε in this context can adopt positive and negativevalues typically between ±0.5. The filtered symbol sequence S_(εVSB)(t)at the output of the lag element 3, which takes the time offset ε·T intoconsideration, is therefore derived according to equation (12):

$\begin{matrix}{{s_{ɛ\;{VSB}}(t)} = {{\mathbb{e}}^{j\frac{2\;\pi}{4T_{VSB}}t} \cdot ( {{\sum\limits_{n = {- \infty}}^{+ \infty}{{a_{R}(n)} \cdot {h_{S}( {t - {ɛ \cdot T_{VSB}} - {2{n \cdot T_{VSB}}}} )}}} + {j \cdot {\sum\limits_{n = {- \infty}}^{+ \infty}{{a_{I}(n)} \cdot {h_{S}( {t - {ɛ \cdot T_{VSB}} - {2{n \cdot T_{VSB}}} - T_{VSB}} )}}}}} )}} & (12)\end{matrix}$

The lag-affected, filtered symbol sequence S_(εVSB)(t) is mixed in a VSBmodulator—modelled as a multiplier 4 in FIG. 1—with a complex carriersignal e^(j(2π(f) ^(T) ^(+Δf)t+Δφ)) to form a VSB-modulatedHF-transmission signal S_(HFVSB)(t). The carrier signal e^(j(2π(f) ^(T)^(+Δf)t+Δφ)) has a carrier frequency f_(T), which has a frequency offsetΔf and phase offset Δφ. The mathematical context for the VSB-modulatedHF transmission signal S_(HFVSB)(t) is presented in equation (13):

$\begin{matrix}{{s_{HFVSB}(t)} = {{\mathbb{e}}^{j\frac{2\;\pi}{4T_{VSB}}t} \cdot \lbrack {{\sum\limits_{n = {- \infty}}^{+ \infty}{{a_{R}(n)} \cdot {h_{S}( {t - {ɛ\; T_{VSB}} - {2{nT}_{VSB}}} )}}} + {j \cdot {\sum\limits_{n = {- \infty}}^{+ \infty}{{a_{I}(n)} \cdot {h_{S}( {t - {ɛ\; T_{VSB}} - {2{nT}_{VSB}} - T_{VSB}} )}}}}} \rbrack \cdot {\mathbb{e}}^{j{({{2\;{\pi{({f_{T} + {\Delta\; f}})}}t} + {\Delta\mspace{11mu}\varphi}})}}}} & (13)\end{matrix}$

Additive, white Gaussian noise (AWGN) n(t), which provides a real and animaginary component n_(R)(t) and n_(I)(t) as shown in equation (14), issuperimposed additively onto the VSB-modulated transmission signalS_(HFVSB)(t) on the transmission path between the transmitter and areceiver.n(t)=n _(R)(t)+j·n _(I)(t)  (14)

The VSB-modulated HF received signal r_(HFVSB)(t) arriving in thereceiver is therefore obtained from equation (15):r _(HFVSB)(t)=s _(HFVSB)(t)+n(t)  (15)

In the receiver, the VSB-modulated received signal r_(HFVSB)(t) withsuperimposed noise n(t) is mixed down into the baseband with the carriersignal e^(−j2πf) ^(T) ^(t) in a demodulator—modelled as the multiplier 5in FIG. 1. The demodulated VSB received signal r_(VSB)(t) at the outputof the demodulator 5 is derived according to the equation (16):

$\begin{matrix}{{r_{VSB}(t)} = {{{{s_{ɛ\;{VSB}}(t)} \cdot {\mathbb{e}}^{j{({{2\;\pi\;\Delta\; f} + {\Delta\;\varphi}})}}} + {n(t)}} = {{{\mathbb{e}}^{j\frac{2\;\pi}{4T_{VSB}}t} \cdot \lbrack {{\sum\limits_{n = {- \infty}}^{+ \infty}{{{a_{R}(n)} \cdot h_{S}}( {t - {ɛ\; T_{VSB}} - {nT}_{VSB}} )}} + {j \cdot {\sum\limits_{n = {- \infty}}^{+ \infty}{{a_{I}(n)} \cdot {h_{S}( {t - {ɛ\; T_{VSB}} - {2{nT}_{VSB}} - T_{VSB}} )}}}}} \rbrack \cdot {\mathbb{e}}^{j{({{2\;\pi\;\Delta\;{ft}} + {\Delta\;\varphi}})}}} + {n(t)}}}} & (16)\end{matrix}$

As can be seen from equation (16), some of the system-theoreticaleffects of the modulator 4 and the demodulator 6 in the transmissionsystem 1 on the VSB-modulated signal are cancelled, so that themodulator 5 and the demodulator 6 in FIG. 1 can be replaced by a singlemultiplier 7 as shown in the reduced block circuit diagram in FIG. 2.

If the VSB baseband received signal r_(VSB)(t) according to equation(16) is mixed with a signal

${\mathbb{e}}^{{- j}\frac{2\;\pi}{4T_{VSB}}t},$if the symbol duration T_(VSB) of the VSB signal according to theequation (17) is adjusted to half the symbol duration T_(S) of aquadrature-modulated signal and if the frequency-displaced cosine filter

$H_{S}( {f - {\frac{1}{4} \cdot f_{SVSB}}} )$according to equation (2) of a quadrature-modulated signal is used asthe transmission filter of the VSB signal, the mathematical relationshipof equation (18) is derived, starting from equation (16), for themodified VSB baseband received signal r_(VSB)′(t).

$\begin{matrix}{T_{VSB} = {\frac{1}{2} \cdot T_{S}}} & (17) \\{{r_{VSB}^{\prime}(t)} = {{( {{\sum\limits_{n = {- \infty}}^{+ \infty}{{a_{R}(n)} \cdot {h_{S}( {t - {ɛ\; T_{S}} - {n \cdot T_{S}}} )}}} + {j \cdot {\sum\limits_{n = {- \infty}}^{+ \infty}{{a_{I}(n)} \cdot {h_{S}( {t - {ɛ\; T_{S}} - {n \cdot T_{S}} - \frac{T_{S}}{2}} )}}}}} ) \cdot {\mathbb{e}}^{{j{({{\Delta\;\text{ft}} + {\Delta\varphi}})}}\;}} + {n(t)}}} & (18)\end{matrix}$

The mathematical term for the modified VSB baseband received signalr_(VSB)′(t) in equation (18) corresponds to the signal display for anoffset QPSK signal, of which the quadrature component is phase-displacedrelative to the in-phase component by half of one symbol length T_(S).

The modified VSB baseband received signal r_(VSB)′(t) is supplied to asampling and holding element 7, referred to below as the second samplingand holding element for an oversampling of the filtered, modified VSBbaseband received signal at a sampling rate f_(A), which is increased bycomparison with the symbol frequency f_(SVSB) of the received signalr_(VSB)′(t) by the oversampling factor os. In this context, theoversampling factor os has a value of 8, as shown in detail in [1]: K.Schmidt: “Digital clock recovery for bandwidth-efficient mobiletelephone systems” [Digitale Taktrückgewinnung für bandbreiteneffizienteMobilfunksysteme], 1994, ISBN 3-18-14 7510-6.

After the sampling of the modified VSB baseband received signalr_(VSB)′(t), an estimation filtering of the signal takes place in anestimation filter 8. The estimation filter 8 has the task of minimisingdata-dependent jitter in the signal. If the transmission filter 2according to equation (2) has a frequency spectrum H_(S)(f), whichcorresponds to a cosine filter with a roll-off factor r, the frequencyspectrum H_(EST)(f) of the estimation filter 8 must be designedaccording to equation (19) dependent upon the frequency spectrumH_(S)(f) of the transmission filter 2 in order to minimisedata-dependent jitter in the modified VSB baseband received signalr_(VSB)′(t), as shown in [1].

$\begin{matrix}{{H_{EST}(f)} = \{ {\begin{matrix}{{H_{S}( {f - f_{S}} )} + {H_{S}( {f + f_{S}} )}} & {{für}{{f{{\leq {\frac{f_{S}}{2}( {1 + r} )}}}}}} \\{beliebig} & {{{für}\frac{f_{S}}{2}( {1 + r} )} < {f} \leq f_{S}} \\{0} & {{{für}\mspace{14mu} f_{S}} < {f}}\end{matrix}\lbrack {{{beliebig} = {random}};{{für} = {for}}} \rbrack} } & (19)\end{matrix}$

The frequency response H_(GES)(f)=H_(S)(f)·H_(EST)(f) of thetransmission system as a whole, consisting of transmission filter 2 andestimation filter 8, can be interpreted according to the equation (20)as a low pass filter H_(GES0)(f) symmetrical to the frequency f=0 with abandwidth of

${\frac{f_{S}}{2} \cdot r},$which is frequency-displaced in each case by

${\pm \frac{f_{S}}{2}}\text{:}$

$\begin{matrix}\begin{matrix}{{H_{GES}(f)} = {{H_{{GES}\; 0}(f)} \star ( {\delta( {f - \frac{f_{S}}{2} + {\delta( {f + \frac{f_{S}}{2}} )}} )} }} \\{= {{H_{{GES}\; 0}( {f - \frac{f_{S}}{2}} )} + {H_{{GES}\; 0}( {f + \frac{f_{S}}{2}} )}}}\end{matrix} & (20)\end{matrix}$

The corresponding impulse response h_(GES)(t) is therefore derivedaccording to equation (21):

$\begin{matrix}{{h_{GES}(t)} = {{{h_{{GES}\; 0}(t)} \cdot ( {{\mathbb{e}}^{{j2}\;\pi\frac{f_{S}}{2}t} + {\mathbb{e}}^{{- {j2}}\;\pi\frac{f_{S}}{2}t}} )} = {{{h_{{GES}\; 0}(t)} \cdot \cos}\;( {2\;\pi\frac{f_{S}}{2}t} )}}} & (21)\end{matrix}$

The signal v_(VSB)′(t) at the output of the estimation filter 8 cantherefore be obtained according to equation (22), in that, in themodified VSB received signal r_(VSB)′(t) in the baseband, as shown inequation (18), the impulse response h_(S)(t) of the transmission filteris replaced with the impulse response h_(GES)(t) of the transmissionsystem as a whole.

$\begin{matrix}{{v_{VSB}^{\prime}(t)} = {{\lbrack {{\sum\limits_{n = {- \infty}}^{+ \infty}{{a_{R}(n)} \cdot {h_{GES}( {t - {ɛ\; T_{S}} - {nT}_{S}} )}}} + {j \cdot {\sum\limits_{n = {- \infty}}^{+ \infty}{{a_{I}(n)} \cdot {h_{GES}( {t - {ɛ\; T_{S}} - \frac{T_{S}}{2} - {nT}_{S}} )}}}}} \rbrack \cdot {\mathbb{e}}^{j{({{2\;\pi\;\Delta\;{ft}} + {\Delta\;\varphi}})}}} + {n(t)}}} & (22)\end{matrix}$

Starting from equation (22), the impulse responseh_(GES)(t−εT_(S)−nT_(S)) can be described according to equation (23):

$\begin{matrix}{{h_{GES}( {t - {ɛ\; T_{S}} - {nT}_{S}} )} = {{{h_{{GES}\; 0}( {t - {ɛ\; T_{S}} - {nT}_{S}} )} \cdot ( {- 1} )^{n} \cdot \cos}\;( {2\;\pi\;\frac{f_{S}}{2}( {t - {ɛ\; T_{S}}} )} )}} & (23)\end{matrix}$

Similarly, the mathematical relationship for the impulse response

$h_{GES}( {t - {ɛ\; T_{S}} - \frac{T_{S}}{2} - {nT}_{S}} )$can be determined in equation (24).

$\begin{matrix}{{h_{GES}( {t - {ɛ\; T_{S}} - \frac{T_{S}}{2} - {nT}_{S}} )} = {{{h_{{GES}\; 0}( {t - {ɛ\; T_{S}} - \frac{T_{S}}{2} - {nT}_{S}} )} \cdot ( {- 1} )^{n} \cdot \sin}\;( {2\;\pi\;\frac{f_{S}}{2}( {t - {ɛ\; T_{S}}} )} )}} & (24)\end{matrix}$

On the basis of the mathematical terms in the equations (23) and (24),the combined terms can be formulated as in equations (25) and (26), andaccordingly, the mathematical context for the output signal v_(VSB)′(t)of the estimation filter 8 in the case of an excitation of thetransmission system 1 with a VSB signal s_(VSB)(t) from equation (22)can be carried over according to equation (27).

$\begin{matrix}{{R(t)} = {\sum\limits_{n = {- \infty}}^{+ \infty}{{a_{R}(n)} \cdot {h_{{GES}\; 0}( {t - {ɛ\; T_{S}} - {nT}_{S}} )} \cdot ( {- 1} )^{n}}}} & (25) \\{{I(t)} = {\sum\limits_{n = {- \infty}}^{+ \infty}{{a_{I}(n)} \cdot {h_{{GES}\; 0}( {t - {ɛ\; T_{S}} - \frac{T_{S}}{2} - {nT}_{S}} )} \cdot ( {- 1} )^{n}}}} & (26) \\{{v_{VSB}^{\prime}(t)} = {\lbrack {{{{R(t)} \cdot \cos}\;( {2\;\pi\;\frac{f_{S}}{2}( {t - {ɛ\; T_{S}}} )} )} + {{j \cdot {I(t)} \cdot \sin}\;( {2\;\pi\;\frac{f_{S}}{2}( {t - {ɛ\; T_{S}}} )} )}} \rbrack \cdot {\mathbb{e}}^{j{({{2\;\pi\;\Delta\;{ft}} + {\Delta\;\varphi}})}}}} & (27)\end{matrix}$

The signal v_(VSB)′(t) at the output of the estimation filter 8according to equation (27) is delayed in a subsequent time-delay unit 9by the timing offset −{circumflex over (ε)}·T_(S). The estimated timingoffset {circumflex over (ε)}, which is determined by an estimation unit,not illustrated here, for estimating the timing offset {circumflex over(ε)} of a VSB-modulated signal, corresponds, in the case of an optimumclock synchronisation, to the actual timing offset ε of the VSBmodulated signal v_(VSB)′(t). In this case, the output signalv_(εVSB)′(t) of the time-delay unit 9 according to equation (28) iscompletely freed from its timing offset.

$\begin{matrix}{{v_{ɛ\;{VSB}}^{\prime}(t)} = {\lbrack {{{{R_{ɛ}(t)} \cdot \cos}\;( {2\;\pi\;{\frac{f_{S}}{2} \cdot t}} )} + {{j \cdot {I_{ɛ}(t)} \cdot \sin}\;( {2\;\pi\;{\frac{f_{S}}{2} \cdot t}} )}} \rbrack \cdot {\mathbb{e}}^{j{({{2\;\pi\;\Delta\;{f \cdot t}} + {\Delta{\;\;}\phi}})}}}} & (28)\end{matrix}$

The associated combined terms R_(ε)(t) and I_(ε)(t) freed from thetiming offset ε·T. are derived according to equations (29) and (30):

$\begin{matrix}{{R_{ɛ}(t)} = {\sum\limits_{n = {- \infty}}^{+ \infty}{{a_{R}(n)} \cdot {h_{{GES}\; 0}( {t - {nT}_{S}} )} \cdot ( {- 1} )^{n}}}} & (29) \\{{I_{ɛ}(t)} = {\sum\limits_{n = {- \infty}}^{+ \infty}{{a_{1}(n)} \cdot {h_{{GES}\; 0}( {t - \frac{T_{S}}{2} - {nT}_{S}} )} \cdot ( {- 1} )^{n}}}} & (30)\end{matrix}$

It is evident from equations (28), (29) and (30) that theclock-synchronised, modified VSB baseband received signal v_(εVSB)′(t)does not provide the time-discrete form according to equation (31)required in order to use the maximum-likelihood method to determine thefrequency-offset and phase-offset estimate Δ{circumflex over (f)} andΔ{circumflex over (φ)}:r(t′)=|r(t′)|·e ^(j(2πΔft′+Δφ))  (31)

According to the invention, the clock-synchronised, modified VSBbaseband received signal v_(εVSB)′(t) is therefore converted, as will beshown below, into a form corresponding to equation (31).

For this purpose, if the output signal v_(VSB)′(t) of the time-delayunit 9 is observed only at the discrete timing points

${t^{\prime} = {{\mu \cdot \frac{T_{S}}{2}}( {{\mu = 0},1,2,{{\ldots\mspace{11mu}{2 \cdot N}} - 1}} )}},$then the output signal v_(εVSB)′(t′) of the time-delay unit 9 iscomposed according to equations (32a), (32b), (32c), and (32d) dependentupon the observed timing point, only of a purely real or purelyimaginary component and a complex rotating phasor e^(j(2πΔf·t′+Δφ)):

$\begin{matrix}{{t^{\prime} = {{{0 \cdot \frac{T_{s}}{2}}\text{:}{v_{ɛ\;{VSB}}^{\prime}( t^{\prime} )}} = {\lbrack {R_{ɛ}( t^{\prime} )} \rbrack \cdot {\mathbb{e}}^{j{({{2\;\pi\;\Delta\;{f \cdot t^{\prime}}} + {\Delta\;\phi}})}}}}}{{v_{ɛ\;{VSB}}^{\prime}(t)} = {{R_{ɛ}(t)} \cdot {\mathbb{e}}^{j{({0\frac{\pi}{2}})}} \cdot {\mathbb{e}}^{j{({{2\;\pi\;\Delta\;{f \cdot t}} + {\Delta\;\phi}})}}}}} & ( {32a} ) \\{{t^{\prime} = {{{1 \cdot \frac{T_{s}}{2}}\text{:}{v_{ɛ\;{VSB}}^{\prime}( t^{\prime} )}} = {\lbrack {j \cdot {I_{ɛ}( t^{\prime} )}} \rbrack \cdot {\mathbb{e}}^{j{({{2\;\pi\;\Delta\;{f \cdot t^{\prime}}} + {\Delta\;\phi}})}}}}}{{v_{ɛ\;{VSB}}^{\prime}( t^{\prime} )} = {{I_{ɛ}( t^{\prime} )} \cdot {\mathbb{e}}^{j{({1\frac{\pi}{2}})}} \cdot {\mathbb{e}}^{j{({{2\;\pi\;\Delta\;{f \cdot t^{\prime}}} + {\Delta\;\phi}})}}}}} & ( {32b} ) \\{{t^{\prime} = {{{2 \cdot \frac{T_{s}}{2}}\text{:}{v_{ɛ\;{VSB}}^{\prime}( t^{\prime} )}} = {\lbrack {- {R_{ɛ}( t^{\prime} )}} \rbrack \cdot {\mathbb{e}}^{j{({{2\;\pi\;\Delta\;{f \cdot t^{\prime}}} + {\Delta\;\phi}})}}}}}{{v_{ɛ\;{VSB}}^{\prime}( t^{\prime} )} = {{R_{ɛ}( t^{\prime} )} \cdot {\mathbb{e}}^{j{({2\frac{\pi}{2}})}} \cdot {\mathbb{e}}^{j{({{2\;\pi\;\Delta\;{f \cdot t^{\prime}}} + {\Delta\;\phi}})}}}}} & ( {32c} ) \\{{t^{\prime} = {{{3 \cdot \frac{T_{s}}{2}}\text{:}{v_{ɛ\;{VSB}}^{\prime}( t^{\prime} )}} = {\lbrack {{- j} \cdot {I_{ɛ}( t^{\prime} )}} \rbrack \cdot {\mathbb{e}}^{j{({{2\;\pi\;\Delta\;{f \cdot t^{\prime}}} + {\Delta\;\phi}})}}}}}{{v_{ɛ\;{VSB}}^{\prime}( t^{\prime} )} = {{I_{ɛ}( t^{\prime} )} \cdot {\mathbb{e}}^{j{({3\frac{\pi}{2}})}} \cdot {\mathbb{e}}^{j{({{2\;\pi\;\Delta\;{f \cdot t^{\prime}}} + {\Delta\;\phi}})}}}}} & ( {32d} ) \\\; & \;\end{matrix}$

The combined terms R_(ε)(t′) and I_(ε)(t′) according to equations (29)and (30) represent real-value low-pass signals, which can be eitherpositive or negative, because of the statistical distribution of thesymbol sequences a_(R)(n) and a_(I)(n).

In the following section, they are described respectively by thetime-dependent real-value amplitude A(t′). Accordingly, for the outputsignal v_(εVSB)′(t′) of the time-delay unit 9 at the individual times

$t^{\prime} = {\mu \cdot \frac{T_{S}}{2}}$ μ = 0, 1, 2, …  2 ⋅ N − 1,instead of timing-related individual equations (32a), (32b), (32c) and(32d), a single mathematical equation (33) containing all of the timingsis obtained for the output signal v_(εVSB)′(t′) of the time-delay unit9:

$\begin{matrix}{{{v_{ɛ\;{VSB}}^{\prime}( t^{\prime} )} = {{A( t^{\prime} )} \cdot {\mathbb{e}}^{j{({\mu\frac{\pi}{2}})}} \cdot {\mathbb{e}}^{j{({{2\;\pi\;\Delta\;{f \cdot t^{\prime}}} + {\Delta\;\phi}})}}}}{{{for}\mspace{14mu} t^{\prime}} = {\mu \cdot \frac{T_{S}}{2}}}} & (33)\end{matrix}$

If the time-discrete output signal v_(εVSB)′(t′) of the time-delay unit9 is frequency-displaced at the individual sampling times

${t^{\prime} = {{{\mu \cdot \frac{T_{S}}{2}}\mspace{31mu}\mu} = 0}},1,2,{{\ldots\mspace{11mu}{2 \cdot N}} - 1}$by a factor

${- \frac{\pi}{2}},$a frequency-displaced, time-discrete signal w_(VSB)′(t′) according toequation (34) is obtained from the time-discrete output signalv_(εVSB)′(t′) of the time-delay unit 9, which, by comparison with thetime-discrete output signal v_(εVSB)′(t′) of the time-delay unit 9, isfreed from the complex term

${\mathbb{e}}^{{j\mu}\frac{\pi}{2}}\text{:}$w _(VSB)′(t′)=A(t′)·e ^(j(2πΔf·t′+Δφ))  (34)

Since the amplitude A(t′) of the signal w_(VSB)′(t′) can adopt positiveand negative values, a modulus for the amplitude must be formed. Amodulus for the amplitude A(t′) of a complex signal is formed bysquaring and subsequent division by the modulus of the complex signal.The phase of the complex signal is doubled by this process, but themodulus remains unchanged.

The use of a squaring of the signal w_(VSB)′(t′) and subsequent divisionby the modulus of the signal w_(VSB)′(t′) leads to the signalx_(VSB)′(t′) according to equation (35), which can be interpreted as atime-discrete, complex rotating phasor with a time-discrete amplitude|A(t′)| and a time discrete phase 2·(2πΔft′+Δφ)=2·(ΔωμT_(S)+Δφ) in thesense of FIG. 4 and which has the form according to equation (31)appropriate for a maximum-likelihood estimation of the frequency offsetand phase offset of the carrier signal:x _(VSB)′(t′)=|A(t′)|·e ^(j2(2πΔf·t′+Δφ)) +n(t′)  (35)

Moreover, in equation (34), the additive interference n(t′) is alsotaken into consideration, which, in a good approximation, isun-correlated and provides a Gaussian distribution. Accordingly, theoptimum estimated value for Δ{circumflex over (f)} and Δ{circumflex over(φ)} is obtained by maximising the maximum-likelihood functionL(Δ{circumflex over (f)}, Δ{circumflex over (φ)}), which according toequation (36), corresponds to a maximising of the real components of alltime-discrete, complex rotating phasors of the signal x_(VSB)′(t′):

$\begin{matrix}{{L( {{\Delta\;\hat{f}},{\Delta\;\hat{\varphi}}} )} = {{Re}\{ {\sum\limits_{\mu}{{{x_{VSB}^{\prime}( {t^{\prime} = {\mu \cdot \frac{T_{s}}{2}}} )}} \cdot {\mathbb{e}}^{{- j}\; 2{({{2\;\pi\;\Delta\hat{f}\;\mu\frac{T_{s}}{2}} + {\Delta\;\hat{\varphi}}}}}}} }} & (36)\end{matrix}$

Maximising the real components of all time-discrete, complex rotatingphasors of the signal x_(VSB)′(t′) can be interpreted as a “turningback” of the time-discrete, complex rotating phasors of the signalx_(VSB)′(t′) respectively by the phase angle

$2 \cdot ( {{2\;\pi\;\Delta\; f\;\mu\frac{T_{s}}{2}} + {\Delta\;\varphi}} )$until these coincide with the real axis in the complex plane.

With reference to the derivation of the mathematical background, adescription of the device according to the invention forcarrier-frequency synchronisation with a VSB-modulated signal accordingto FIG. 3 and of the method according to the invention forcarrier-frequency synchronisation with a VSB-modulated signal accordingto FIG. 5 is provided below.

In the case of an inverted position of the sideband, the VSB basebandreceived signal r_(VSB)(t) according to equation (16) is subjected, in aunit for sideband mirroring 10 in the device according to the inventionas shown in FIG. 3, to a mirroring of the sideband at the carrierfrequency f_(T) into the normal position.

Following this, the VSB baseband received signal r_(VSB)(t) is mixeddown in a down mixer 11 by means of a mixer signal

${\mathbb{e}}^{{- j}\frac{2\;\pi}{4T_{VSB}}t}$by the frequency

$\frac{f_{SVSB}}{4}$into a modified VSB baseband received signal r_(VSB)′(t) according toequation (18).

A downstream sampling and holding element 7 with an oversampling factoros samples the modified VSB the baseband received signal r_(VSB)′(t). Anestimation filtering in the sense of equation (22) or respectively (27)also takes place in a signal-adapted estimation filter 8. A clocksynchronisation of the output signal v_(VSB)′(t) of the estimationfilter 8 by the timing offset −{circumflex over (ε)}·T_(S) is carriedout in a downstream time-delay unit 9 according to equation (28). Theestimated timing offset {circumflex over (ε)}, which is determined by anestimation unit, not illustrated here, for the estimation of the timingoffset {circumflex over (ε)} of a VSB-modulated received signal,corresponds, in the case of an optimum clock synchronisation, to theactual timing offset ε of the VSB-modulated baseband received signalr_(VSB)′(t).

The clock-synchronised output signal v_(εVSB)′(t) of the time-delay unit9 is sampled down in a sampling and holding element 12 referred to belowas the first sampling and holding element to two sampling values persymbol period T_(S).

The output signal v_(εVSB)′(t′) of the first sampling and holdingelement 12 is supplied to a complex multiplier 13, in which it issubjected to a sampling-time-related phase offset by the phase angle

${- \mu} \cdot {\frac{T_{S}}{2}.}$

The output signal w_(VSB)′(t′) of the complex multiplier 13, accordinglyphase-displaced in its phase relative to the signal v_(εVSB)′(t′), issupplied to a unit for modulus-scaled squaring 14, consisting of asquaring unit, a modulus former and a divider connected downstream ofthe squaring unit and the modulus former, in which a modulus for itsamplitude is formed and its phase is doubled.

The signal at the output of the unit for modulus-scaled squaring 14represents the modified VSB baseband received signal x_(VSB)′(t′), whichthe signal-processing unit 15 has generated from the clock-synchronisedVSB baseband received signal v_(εVSB)′(t) by undersampling in the firstsampling and holding element 12, by phase displacement in the complexmultiplier 13 and by modulus formation of the amplitude or respectivelydoubling of the phase in the unit for modulus-scaled squaring 14.

The estimated values Δ{circumflex over (f)} and Δ{circumflex over (φ)}for the frequency offset and phase offset of the carrier signal aredetermined, as described, for example, in DE 103 09 262 A1, from thetime-discrete, modified VSB a baseband received signal x_(VSB)′(t′) in asubsequent maximum-likelihood estimator 18.

A frequency offset and phase offset estimator, such as that disclosed inDE 103 09 262, which avoids any 2π slips occurring in the phasecharacteristic—so-called “cycle slips”, which, in the case of a phaseregression, result, through small amplitudes of the time-discrete,modified received signal x(t′), from the superimposed interference, canbe used as a maximum-likelihood estimator. Accordingly, the phaseregression cannot be used for this application.

The method according to the invention for carrier-frequencysynchronisation of a VSB-modulated signal is described below withreference to FIG. 5.

In procedural stage S10, the sideband of the VSB baseband receivedsignal r_(VSB)(t) is mirrored by the carrier frequency f_(T) from aninverted position into a normal position, if the sideband is disposed inthe inverted position.

In the next procedural stage S20, the VSB baseband received signalr_(VSB)(t) is mixed down with a mixer signal

${\mathbb{e}}^{{- j}\frac{2\;\pi}{4T_{VSB}}t}$by the frequency

$\frac{f_{SVSB}}{4}$into a modified baseband received signal r_(VSB)′(t′) according toequation (18).

In the next procedural stage S30, the modified VSB baseband receivedsignal r_(VSB)′(t) is sampled in a second sampling with an oversamplingfactor of typically eight.

The sampled, modified VSB baseband received signal e_(VSB)′(t′) issupplied, in procedural stage S40, to an estimation filter according toequations (22) and (27) respectively, which minimises data-dependentjitter in the sampled, modified baseband received signal e_(VSB)′(t).

In the next procedural stage S50, a clock-synchronisation of the sampledand filtered, modified VSB baseband received signal v_(εVSB)′(t) takesplace according to equation (28) by means of a time-delay unit 9, whichreceives the estimated timing offset {circumflex over (ε)} from anestimator, which is not described in greater detail here.

In the next procedural stage S60, an additional sampling takes place—afirst sampling—of the clock-synchronised VSB baseband received signalv_(εVSB)′(t) at two sampling values per symbol period T_(S) according toequation (33).

In the next procedural stage S70, the additionally sampledclock-synchronised VSB baseband received signal v_(εVSB)′(t′) isfrequency displaced by complex multiplication with asampling-time-related multiplication factor

${\mathbb{e}}^{{- j}\;\mu\frac{\pi}{2}}$to compensate the respective inverse complex factor

${\mathbb{e}}^{j\;\mu\frac{\pi}{2}}$in the additionally sampled signal v_(εVSB)′(t′) according to equation34.

The next procedural stage S80 contains the modulus formation of thetime-discrete amplitudes A(t′) and a doubling of the time-discretephases 2πΔft′+Δφ of the frequency-displaced, additionally-sampled andclock-synchronised VSB baseband received signal w_(VSB)′(t′) accordingto equation (35).

In the next procedural stage S90, the time-discrete, modified VSBbaseband received signal x_(VSB)′(t′) obtained from theclock-synchronised VSB baseband received signal v_(εVSB)′(t) inprocedural stages S60, S70 and S80 by means of the signal-processingunit 15 is used to determine the frequency offset and phase offset valueΔ{circumflex over (f)} and Δ{circumflex over (φ)} of the carrier signalby means of maximum-likelihood estimation according to equation (36).The maximum-likelihood estimator used should ideally be able to overcomeany phase slips—so-called “cycle slips” resulting from interferencesignals, which are superimposed on the modified VSB baseband receivedsignal v_(εVSB)′(t) in the case of small amplitudes of the modified VSBbaseband received signal v_(εVSB)′(t), and is disclosed, for example, inDE 103 09 262 A1.

1. A computer-implemented method for carrier-frequency synchronizationof a carrier signal (e^(j(2π(f) ^(T) ^(+Δf)t+Δφ))) affected by afrequency offset or a phase offset (Δf, Δφ) comprising: receiving avestigial-sideband-modulated signal (r_(VSB)(t)); converting thereceived signal into a modified vestigial-sideband-modulated receivedsignal (x_(VSB)′(t′)) with time-discrete, complex rotating phasors(|A(t′)|·e^(j2(2πΔf·t′+Δφ))), in which only the time-discrete phases(2·(2πΔft′+Δφ)) are dependent upon the frequency offset or phase offset(Δf, Δφ); and performing a maximum-likelihood estimation of thefrequency and phase offsets using the time-discrete, complex rotatingphasors, wherein the time-discrete phases (2·(2πΔft′+Δφ)) of the complexrotating phasors (|A(t′)|·e^(j2(2πΔf·t′+Δφ))), of the modifiedvestigial-sideband-modulated received signal (x_(VSB)′(t′)), aredependent only upon the frequency offset or phase offset (Δf, Δφ), andthe symbol duration (T_(VSB)) of the vestigial-sideband-modulatedreceived signal (r_(VSB)(t)) is half of the symbol duration (T_(S)) ofthe received signal (r(t′)).
 2. A computer-implemented method forcarrier-frequency synchronization of a carrier signal (e^(j(2π(f) ^(T)^(+Δf)t+Δφ))) affected by a frequency offset or a phase offset (Δf, Δφ)comprising: receiving a vestigial-sideband-modulated signal(r_(VSB)(t)); converting the received signal into a modifiedvestigial-sideband-modulated received signal (x_(VSB)′(t′)) withtime-discrete, complex rotating phasors (|A(t′)|·e^(j2(2πΔf·t′+Δφ))), inwhich only the time-discrete phases (2·(2πΔft′+Δφ)) are dependent uponthe frequency offset or phase offset (Δf, Δφ); and performing amaximum-likelihood estimation of the frequency and phase offsets usingthe time-discrete, complex rotating phasors, wherein the conversion ofthe vestigial-sideband-modulated received signal (r_(VSB)(t)) consistsof a down mixing by a quarter of the symbol frequency$( \frac{f_{SVSB}}{4} ),$  a first sampling at two samplingvalues per symbol period (T_(S)), a complex multiplication and amodulus-scaled squaring.
 3. A computer-implemented method for clocksynchronization of a carrier signal (e^(j(2π(f) ^(T) ^(+Δf)t+Δφ)))affected by a frequency offset or a phase offset (Δf, Δφ) comprising:receiving a vestigial-sideband-modulated signal (r_(VSB)(t)); convertingthe received signal into a modified vestigial-sideband-modulatedreceived signal (X_(VSB)′(t′)) with time-discrete, complex rotatingphasors (|A(t′)|·e^(j2(2πΔf·t′+Δφ))), in which only the time-discretephases (2·(2πΔft′+Δφ)) are dependent upon the frequency offset or phaseoffset (Δf, Δφ); and performing a maximum-likelihood estimation of thefrequency and phase offsets using the time-discrete, complex rotatingphasors, wherein the symbol duration (T_(VSB)) of thevestigial-sideband-modulated received signal (r_(VSB)(t)) is half of thesymbol duration (T_(S)) of the received signal (r(t′)).
 4. Acomputer-implemented method for clock synchronization according to claim2, wherein the complex multiplication takes place with the complex phaseangle ${\mathbb{e}}^{{- j}\;\mu\frac{\pi}{2}},$ wherein μ is thesampling index.
 5. A computer-implemented method for clocksynchronization according to claim 2, wherein the modulus-scaledsquaring takes place by parallel squaring, modulus formation andsubsequent division.
 6. A computer-implemented method for clocksynchronization according to claim 2, wherein in the case of an invertedposition of the sideband of the vestigial-sideband-modulated receivedsignal (r_(VSB)(t)), the down mixing is preceded by a mirroring of thesideband from the inverted position into the normal position.
 7. Acomputer-implemented method for clock synchronization according to claim2, wherein the down mixing of the vestigial-sideband-modulated receivedsignal (r_(VSB)(t)) is followed by an over-sampling, an estimationfiltering and a clock synchronization.
 8. A computer-implemented methodfor clock synchronization according to claim 2, wherein the conversionof the vestigial-sideband-modulated received signal (r_(VSB)(t)) isfollowed by a maximum-likelihood estimation of the frequency offset andphase offset (Δf, Δφ) of the carrier signal (e^(j(2π(f) ^(T)^(+Δf)t+Δφ))).
 9. A device for carrier-frequency synchronization of acarrier signal (e^(j(2π(f) ^(T) ^(+Δf)t+Δφ))) affected by a frequencyoffset or phase offset (Δf, Δφ) with a maximum-likelihood estimator forestimating the frequency offset or phase offset (Δf, Δφ) of the carriersignal (e^(j(2π(f) ^(T) ^(+Δf)t+Δφ))) from a received signal (r(t′))with time-discrete, complex rotating phasors (|r(t′)|·e^(j2πΔf·t′+Δφ)),in which only the time-discrete phases (2πΔft′+Δφ) are dependent uponthe frequency offset or phase offset (Δf, Δφ), wherein themaximum-likelihood estimator is preceded by a signal-processing unit anda down mixer, which converts the received signal (r(t′)) which is avestigial-sideband-modulated received signal (r_(VSB)(t)), into amodified vestigial-sideband-modulated received signal (X_(VSB)′(t′))with time-discrete complex rotating phasors(|A(t′)|·e^(j2(2πΔf·t′+Δφ))), in which only the time-discrete phases(2πΔft′+Δφ) are dependent upon the frequency offset or phase offset (Δf,Δφ), and wherein the signal-processing unit includes of a first samplingunit, a complex multiplier and a unit for modulus-scaled squaring,wherein the unit for modulus-scaled squaring includes a squaring elementand a parallel-connected modulus former with a divider connecteddownstream of the squaring element and the modulus former.
 10. A devicefor carrier-frequency synchronization of a carrier signal (e^(j(2π(f)^(T) ^(+Δf)t+Δφ))) affected by a frequency offset or phase offset (Δf,Δφ) with a maximum-likelihood estimator for estimating the frequencyoffset or phase offset (Δf, Δφ) of the carrier signal (e^(j(2π(f) ^(T)^(+Δf)t+Δφ))) from a received signal (r(t′)) with time-discrete, complexrotating phasors (|r(t′)|·e^(j2πΔf·t′+Δφ)), in which only thetime-discrete phases (2πΔft′+Δφ) are dependent upon the frequency offsetor phase offset (Δf, Δφ), wherein the maximum-likelihood estimator ispreceded by a signal-processing unit and a down mixer, which convertsthe received signal (r(t′)) which is a vestigial-sideband-modulatedreceived signal (r_(VSB)(t)), into a modifiedvestigial-sideband-modulated received signal (X_(VSB)′(t′)) withtime-discrete complex rotating phasors (|A(t′)|·e^(j2(2πΔf·t′+Δφ))), inwhich only the time-discrete phases (2πΔft′+Δφ) are dependent upon thefrequency offset or phase offset (Δf, Δφ), and wherein the down mixer ispreceded by a unit for sideband mirroring and is followed by a secondsampling unit, an estimation filter and a time-delay unit for clocksynchronization.
 11. A device for carrier-frequency synchronization of acarrier signal (e^(j(2π(f) ^(T) ^(+Δf)t+Δφ))) affected by a frequencyoffset or phase offset (Δf, Δφ) with a maximum-likelihood estimator forestimating the frequency offset or phase offset (Δf, Δφ) of the carriersignal (e^(j(2π(f) ^(T) ^(+Δf)t+Δφ))) from a received signal (r(t′))with time-discrete, complex rotating phasors (|r(t′)|·e^(j2πΔf·t′+Δφ)),in which only the time-discrete phases (2πΔft′+Δφ) are dependent uponthe frequency offset or phase offset (Δf, Δφ), wherein themaximum-likelihood estimator is preceded by a signal-processing unit anda down mixer, which converts the received signal (r(t′)) which is avestigial-sideband-modulated received signal (r_(VSB)(t)), into amodified vestigial-sideband-modulated received signal (X_(VSB)′(t′))with time-discrete complex rotating phasors(|A(t′)|·e^(j2(2πΔf·t′+Δφ))), in which only the time-discrete phases(2πΔft′+Δφ) are dependent upon the frequency offset or phase offset (Δf,Δφ), and wherein the down mixer is followed by a second sampling unit,an estimation filter and a time-delay unit for clock synchronization.12. A computer-readable storage medium with electronically-readablecontrol signals, which can cooperate with a programmable computer ordigital signal processor, the storage medium being encoded withinstructions for executing the method according to claim
 1. 13. Acomputer software product with program-code means stored on amachine-readable storage medium, in order to implement all of the stagesaccording to claim 1, when the program is executed on a computer or adigital signal processor.
 14. A computer-readable medium embodyingprogram code means, the program code means comprising instructions forimplementing all of the stages according to claim 1, when the program isexecuted on a computer or a digital signal processor.
 15. Acomputer-readable medium embodying program code means, in order toimplement all of the stages according to claim 1, when the program isstored on a machine-readable data storage medium.